I have the same issue with my gf. We’ve been together for about 4 months, she’s a great lady and I love most things about her, except for her weight. I’ve told her directly I do not like it and she got very upset with me so I decided not to do that again.
She has a genuine desire to be with me so I take her hiking as often as I can. I deliberately push her and she sees how easy it is for me when compared to her. She then raises the subject of getting fit and losing weight. The subject is much easier when she raises it and I can talk to her more about it that way. This seems to be the way to go, do things with them that show them how unfit they are and get them talking about it.
I also tell her about my activities, I tell her every time I go to the gym. If I put a few pounds on I tell her and tell her how much I don’t want to put weight on. This lead by example seems to get her attention.
My gf does seem to be making an effort. Whether she actually looses weight remains to be seen.